New Results on the Sum of Two Generalized Gaussian Random

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α Thereby the distribution of the sum, that appears also as the Alternative expressions of cos(x) and ex in terms of the FHF convolution of the single distribution, is needed in this seismic are available in [14, Eq. (2.9.8) & Eq. (2.9.4)] model. Moreover, in communications, it is shown above that x2 in some instances the noise and the multi user interference can cos(x)= √π H1,0 (7) 0,2 4 (0, 1), ( 1 , 1) be modeled as GG white noise [1], [12]. Therefore, the total 2 perturbations at reception, deﬁned as the sum of noise and α 1 e−x = H1,0 x (8) interference, is modeled as the sum the GG signals. Many α 0,1 (0, 1 ) α other applications can be found in the literature to motivate the study of sum distribution of GG signals. Hence the integral identity deﬁned in [14, Eq. (2.8.4)], solves Per consequence, it is important to study the statistics and the integral of the product of two FHF function over the the density of the SGG distribution. The PDF was not derived positive real numbers. As a consequence the CF can be re- before and so the CDF. Actually, the approach used in this written as work is based on the CF which was investigated in [13] for √πΛ ∞ t2 ϕ (t)= eitµ H1,0 x2 α > 1. However in our case we are studying the CF for any α Γ(1/α) 0,2 4 (0, 1), ( 1 , 1) × Z0 2 value of α using another approach of calculation based on the properties of the Fox H function (FHF) [14]. From the H1,0 Λx dx 0,1 (0 , 1 ) CF of one GGRV, we get the CF of the sum. Hence, the α relation between the CF and the distribution densities leads us √π σ2Γ(1/α ) (1 1 , 2 ) = eitµH1,1 t2 − α α . to investigate the PDF and the CDF of such distribution and 1,2 1 Γ(1/α) " 4Γ(3/α) (0, 1), ( , 1) # 2 its statistics (moments, cumulant, kurtosis...). (9)

B. Organization of the Paper Note that the result proved in (9) is valid for all positive The remaining of this paper is organized as follows. In shape parameter α> 0. A previous demonstration was derived Section II, we investigate the statistics of the GGD, such as the in [13] but only for α > 1, while the authors provided CF, the moments, cumulant, and kurtosis. In Section III, the an expression of the CF of GGD in terms of the Fox- PDF and some statistics of the SGG distribution are presented. Wright generalized hypergeometric function pΨq( ), which Next in Section IV, we analyze a method to approximate the is a special case of the FHF [14, Eq. (2.9.29)].· The CF PDF of the sum by a PDF of one GGD and the performance expression and derivation presented in this work present one of such approximation is studied. Finally Section V concludes of the contributions of the paper. the paper.

B. Moment Generating Function II. GENERALIZED GAUSSIAN DISTRIBUTION STATISTICS The moment generating function (MGF) can be directly A. Characteristic Function concluded from the CF by the relation Mα(t) = ϕα( it), Let α > 0, and X be random variable (RV) following a so the MGF is obtained by − GGD(µ, σ, α) tµ 2 (1 1 , 2 ) E itX √πe 1,1 σ Γ(1/α) 2 α α Theorem 1. The CF of X, [e ], is given by Mα(t)= H1,2 − t − 1 . Γ(1/α) " 4Γ(3/α) (0, 1), ( 2 , 1) # 2 1 2 (10) √π itµ 1,1 σ Γ(1/α) 2 (1 α , α ) ϕα(t)= e H1,2 t − 1 , In some special cases, the MGF can be expressed in terms Γ(1/α) " 4Γ(3/α) (0, 1), ( , 1) # 2 of elementary functions. For example for Gaussian case, i.e. (4) α = 2, and using the special case of the FHF in [14, Eq. ·,· where H· ·[ ] is the Fox H function (FHF) [14, Eq. (1.1.1)], (2.9.4)], the MGF of Gaussian is , · [15]. 2 1,0 σ 2 + 1 2 2 Proof: Let us start from the deﬁnition of the CF and the M (t)= etµH − t = etµ 2 σ t (11) 2 0,1 2 (0, 1) PDF of the GGD. In fact the CF can be expressed in integral form as Another special case appears interesting is the Laplacian

itX itx distribution (i.e. α = 1). Using the identity [14, Eq. (2.9.5)], ϕα(t)= E[e ]= e fX (x)dx R the MGF of Laplacian distribution is given by Z αΛ itµ itx α α 2 (5) tµ 1,1 σ 2 (0, 2) = e e exp( Λ x ) dx. M (t)= √πe H − t 2Γ(1/α) R − | | 1 1,2 8 (0, 1), ( 1 , 1) Z 2 Since x is an even function, the integral in (5) is the cosine 1 (0 , 1) | | = etµH1,1 σ2t2 transform of the exponential component 1,1 −2 (0 , 1) ∞ etµ αΛ itµ α ϕα(t)= e cos(tx)exp( (Λx) ) dx. (6) = 1 . (12) Γ(1/α) − 1 σ2t2 Z0 − 2 3

C. Moments and Cumulant Another statistics appear interesting to evaluate, in occurrence Without loss of generality, we are focusing our study to zero the kurtosis. The kurtosis, Kurt(X), is defined as the fourth mean random variables (i.e. µ =0). cumulant divided by the square of the second cumulant of the Due to the symmetry of the PDF of GGD, the odd moments distribution. In the GGD case the kurtosis is equal to vanish and the even moments obtained as follows k4(X) Γ(1/α)Γ(5/α) 2n+1 (18) n Γ( ) Kurt(X)= 2 = 2 3. E 2n 2n Γ(1/α) α k2(X) Γ(3/α) − m2n(X) = [X ] = σ Γ(3/α)n Γ(1/α) E 2n+1 ( m2n+1(X) = [X ] =0 One can easily check that (18) confirms that the Gaussian (13) kurtosis is equal to 0 and the Laplacian kurtosis is equal to 3. Once the MGF and the moments are obtained, one can At this stage, the statistics of one GGRV are expressed investigate the expression of the cumulant generating function in closed form. The next section considers the densities and (CGF) and the cumulant of the GGD. Actually the CGF, statistics of the SGG distribution. KX (t) (or Kα(t)), is defined as

Kα(t) = log Mα(t) III. SUM OF TWO INDEPENDENT GG RANDOM VARIABLES √π As known the CF of the sum of two independent RV is the = log( ) Γ(1/α) product of their CFs. Since the CF of the GGD is deﬁned in the previous section, the CF of the sum can be easily obtained σ2Γ(1/α) (1 1 , 2 ) 1,1 2 α α and so the densities by inverse Laplace transform of the CF. + log H1,2 − t − 1 . " 4Γ(3/α) (0, 1), ( 2 , 1) # In fact, let and X GGD(µ1, σ1, α) Y GGD(µ2, σ2,β) (14) two independent∼ random variables following∼ a GGD, and let

Z = X + Y be their sum. It is clear that the first and second By definition, the n-th cumulant, noted kn(X), is the n-th moment of Z are easy to find term in the Taylor series expansion of Kα(t) at 0. E Theorem 2. The even cumulants of a zero mean GG random [Z]= µ = µ1 + µ2 variable can be expressed in terms of the even moments of E 2 2 2 2 X [(Z µ) ]= σ = σ1 + σ2 . (19) X by −

(2n)!(m1 + + mn 1)! A. PDF and CDF of the Sum of Two GGRV k2n(X)= ··· − − m1!m2! ...mn! × m1+2m2+...nmn=n The PDF of a random variable is known as the inverse X mj σ2j Γ(1/α)jΓ( 2j+1 ) Laplace transform of the CF. The CF of Z is given by α , (15) −Γ(3/α)jΓ(1/α)(2j)! 1≤j≤n ! ϕZ (t)= ϕX (t)ϕY (t) Y itµ 2 1 2 and the odd cumulants are equal to zero. πe 1,1 σ1 Γ(1/α) 2 (1 α , α ) = H1,2 t − 1 Γ(1/α)Γ(1/β) " 4Γ(3/α) (0, 1), ( 2 , 1) # × Proof: A cumulant kn(X) is the n-th derivative of the n n d Kα(t) d Kα(0) 2 1 2 CGF evaluated at zero, k (X) = n = n . σ Γ(1/β) (1 , ) n dt dt 1,1 2 2 − β β t=0 H1,2 t 1 . Since the CGF appears as the composite of two functions, we " 4Γ(3/β) (0, 1), ( , 1) # 2 may use the Fa`adi Bruno’s formula [16, Eq. (2)] that computes (20) the n-th derivative of composite functions By applying the Laplace transform inverse to (20), the PDF n d Kα(t) n!(m1 + + mn 1)! of Z is given by the following theorem. n = ··· − dt − m1!m2! ...mn! × m1,...,mn Theorem 3. The PDF of the sum of two independent GG X mj M (j)(t) random variable can be expressed in terms of the bivariate α , (16) −j!M (t) FHF [17] in (21). ≤ ≤ α ! 1 Yj n n The FHF of two variables [17], also known as the Bivariate Fox H-function (BFHF) H· ·;· ·;· ·[ , ] is a generalization of the sum is over m1,m2,...,mn such that jmj = n. · ·;· ·;· · the FHF. Its MATLAB implementation· · is outlined in [18]. j=1 2 2 Thereby evaluating (16) at zero and replacingX the moment by σ1 Γ(1/α) σ2 Γ(1/β) Proof: Let A = 4Γ(3/α) and B = 4Γ(3/β) . The inverse its expression, one get the ﬁnal expression of the cumulant. Laplace transform of the CF (20) of Z gives the PDF The cumulant of low order are easy to expressed 1 −itz 2 fZ (z)= e ϕZ (t)dt k2(X)= σ 2π R Z 1 2 4 Γ(1/α)Γ(5/α) 1 (1 , ) k4(X)= σ 3 1,1 2 α α Γ(3/α)2 − = H1,2 A t − 1 2Γ(1/α)Γ(1/β) R " (0, 1), ( 2 , 1) # × Γ(1/α)2Γ(7/α) Γ(1/α)Γ(5/α) Z 6 1 2 k6(X)= σ 3 15 2 + 30 . 1,1 (1 , ) Γ(3/α) − Γ(3/α) H B t2 − β β eit(µ−z)dt (23) 1,2 1 (17) " (0, 1), ( , 1) # 2

√π f (z)= Z Γ(1/α)Γ(1/β) z µ × | − | σ2Γ(1/α) σ2Γ(1/β) ( 1 , 1, 1), (0, 1, 1) (1 1 , 2 ) (1 1 , 2 ) 0,1;1,1;1,1 1 2 2 α α − β β (21) H2,0;1,2;1,2 2 , 2 − 1 1 " Γ(3/α)(z µ) Γ(3/β)(z µ) (0, 1), ( , 1) (0, 1), ( , 1) # − − 2 2

1 √π sign(z µ) F (z)= + − Z 2 2Γ(1/α)Γ(1/β) × σ2Γ(1/α) σ2Γ(1/β) ( 1 , 1, 1), (1, 1, 1) (1 1 , 2 ) (1 1 , 2 ) 0,1;1,1;1,1 1 2 2 − α α − β β (22) H2,0;1,2;1,2 2 , 2 1 1 "Γ(3/α)(z µ) Γ(3/β)(z µ) (0, 1), ( , 1) (0, 1), ( , 1) # − − 2 2

The first two FHF are even functions, so the integral becomes 2) Moment: The moments of Z can be obtained from the a cosine transform of the product of these two FHF functions binomial formula that describes the integer power of the sum of two numbers. Hence known the moment of X and Y (13), it ∞ 1 2 1 1,1 2 (1 α , α ) appears that the odd moments of Z, m2n+1(Z), vanish while fZ (z)= H A t − 2Γ(1/α)Γ(1/β) 1,2 (0, 1), ( 1 , 1) × the even moments are given by Z0 " 2 # k (1 1 , 2 ) σ2nΓ( 1 )n n 2 Γ( 1 )Γ( 3 ) 1,1 2 − β β 2 β 2n σ1 α β H1,2 B t 1 cos(t(µ z)) dt m2n(Z)= " (0, 1), ( , 1) # − Γ( 1 )Γ( 1 )Γ( 3 )n 2k σ2 Γ( 3 )Γ( 1 ) × 2 α β β k=0 2 α β ! (24) X 2k +1 2n 2k +1 Γ Γ − (26) α β As seen before, the cosine has a representation in terms of the FHF (7). So we are facing an integral that involves the product 3) Cumulant and Kurtosis: From the MGF, it is easier to of three FHFs over the positive real numbers. Such integral is get the CGF by applying the logarithm to the MGF MZ (t). solved in [17, Eq. (2.3)] and it is expressed in terms of the Thereby the CGF of Z is the sum of the CGF of X and the BFHF, which give us the final expression of the PDF of Z. CGF of Y , KZ (t)= KX (t)+KY (t). Moreover, the cumulant The CDF of Z is the primitive of fZ that vanishes at ( ). of Z is expressed also as the sum of the cumulant of X and −∞ Back to (24), it appears that the CDF is expressed in term of an the cumulant of Y . Note that the odd cumulant are equal to integral involving the product of two FHFs and sine function. zero while the even ones are given by The latter can be expressed in terms of the FHF for positive argument [14, Eq. (2.9.7)]. Thereby, the CDF of the sum of k2n(Z)= k2n(X)+ k2n(Y ) two independent GGRV becomes the integral of the product (2n)!(m1 + + mn 1)! = ··· − of three FHFs which is evaluated in terms of the BFHF. − m !m ! ...m ! × m +2m +...nm =n 1 2 n 1 2X n Corollary 3.1.The CDF, F (z), of the SGG distribution is mj Z σ2j Γ( 1 )j Γ( 2j+1 ) given in (22). 1 α α  − Γ( 3 )j Γ( 1 )(2j)! ≤ ≤ α α ! In (22), sign(x) gives the sign of the real number x. The 1 Yj n results in Theorem.3 and Corollary 3.1 represent new results  2k 1 k 2k+1 mk σ2 Γ( ) Γ( ) and they were not investigated before, which make them the + β β . (27) − Γ( 3 )kΓ( 1 )(2k)!  essential contribution of this paper. ≤ ≤ β β ! 1 Yk n Once the cumulant expression is evaluated, the kurtosis of Z B. Statistics of Z can be expressed, per definition, in terms of the fourth and second moments as In the following analysis, the zero mean case is considered, k4(Z) k4(X)+ k4(Y ) while the non zero mean random variable can be obtained from Kurt(Z)= 2 = 2 . k2(Z) (k2(X)+ k2(Y )) the zero mean random variable by a simple shift Z = Z0 + µ. 2 2 2 1) MGF: As mentioned before the MGF of Z can be Note that k2(X)+ k2(Y ) = σ1 + σ2 = σ . Thus a relation obtained from the CF by the relation MZ (t)= ϕZ ( it) which between the kurtosis of Z, X and Y appears as gives the MGF of Z as − σ2 σ2 Kurt(Z)= 1 Kurt(X)+ 2 Kurt(Y ). (28) 2 1 2 σ2 σ2 π 1,1 σ1 Γ(1/α) 2 (1 α , α ) MZ (t)= H1,2 − t − 1 The final expression of the kurtosis of Z is thus given by Γ(1/α)Γ(1/β) " 4Γ(3/α) (0, 1), ( , 1) # 2 2 1 2 Kurt(Z)= 1,1 σ2Γ(1/β) 2 (1 β , β ) H1,2 − t − . 4 1 5 4 1 5 2 2 4Γ(3/β) 1 σ Γ( )Γ( ) σ Γ( β )Γ( β ) σ σ " (0, 1), ( 2 , 1) # 1 α α 2 1 2 (29) 4 3 + 4 3 +6 4 3 (25) σ Γ( )2 σ Γ( )2 σ − α β

IV. APPROXIMATION OF THE PDF OF THE SUM OF TWO Lets define the ratio between the variance of and as 2 X Y σ1 GGRV δ = 2 , so the equation on γ can be written in terms of δ as σ2 The expression of the PDF of the sum of two independent 1 5 1 5 Γ( )Γ( ) 1 Γ( 1 )Γ( 5 ) Γ( )Γ( ) GGRV (21) is quite high complex since it is expressed in γ γ = δ2 α α + β β +6δ , 3 2 2 3 2 3 2 terms of the BFHF. Therefore, an approximation of the PDF Γ( γ ) (1 + δ) Γ( α ) Γ( β ) ! is highly recommended to simplify the calculations and study, (31) in simple way, the performance of systems in which the PDF By knowing α, β, and δ, (30) is written as h(γ)= C, where of the sum is needed, like, for example, the evaluation of the h(γ) is a function on γ, and C is a known positive constant. In symbol error rate (SER) of an M-phase shift keying (MPSK) Fig. 1, the function h is drawn versus γ to analyze its behavior. over an GGN channel. Such analysis needs the PDF and the Therefore, it appears that the function h( ) is a bijection. As · CDF of the SGG distribution. such the equation h(γ) = C has only one solution in the In this section we are investigating the approximation of the positive real axis. Which mean that γ exists and it is unique. PDF of Z by the PDF of another GG random variable with The value of γ is given as γKurt in Table I for some scenarios shape factor γ to be determined. In [10], it has been proved that along with other values of γ obtained from other approaches the PDF of the sum cannot be a PDF of one GGRV. However that we will discuss later on. the authors proved that both PDFs have the same properties (symmetric, convexity, monotonicity...). Furthermore, the PDF B. Best Tail Approximation of the sum of two i.i.d. GGRV was approximated by the PDF of GGD. From that analysis, an approximation of the PDF of Another method to estimate γ consist of taking the best Z by the PDF of GGD is needed and worth pursuing. choice of γ that minimizes the square error of the tail. In As shown in (1), 3 parameters are needed to characterize a other words γ is chosen so the error between the exact PDF, GGD, namely, the mean, the variance and the shape factor. The fZ (z) and the approximated PDF fγ(z) at the tail is minimal. The tail is defined so z is above some level z nσ mean and the variance are given in (19). Therefore, we need ≥ ∞ to find a method to get the shape factor γ. In what follows, 2 γTail = arg min (fγ(z) fZ (z)) dz, (32) three approaches are presented. γ>0 − Znσ where n is chosen to define the desired region of the tail of A. Kurtosis Approach the distribution. The minimization in (32) cannot be solved The first method to estimate γ is by using the kurtosis of the analytically by the available tools since it contains a shifted distributions. Since the kurtosis of the sum is already known, integral of a BFHF and FHF which is not known yet. A the shape factor can be obtained by equalizing both kurtosis. numerical evaluation of γTail is given in Table I for different Thereby, we get the following equation to solve values of δ.

Kurt(Zγ )= Kurt(Z) C. CDF Approximation 1 5 ⇐⇒ 1 5 Γ( )Γ( ) σ4 Γ( 1 )Γ( 5 ) σ4 Γ( )Γ( ) σ2σ2 This method is used to obtain the shape parameter that γ γ = 1 α α + 2 β β +6 1 2 , 3 2 4 3 2 4 3 2 4 minimizes the error between the CDF of Z and the approx- Γ( γ ) σ Γ( α ) σ Γ( β ) σ (30) imated CDF. Such approximation will give an asymptotic while Zγ GGD(µ,σ,γ) is the approximated RV of Z with approximation of the complementary CDF (CCDF) which parameter∼γ. is needed in the computation of the probability of error. Mathematically, the shape parameter is given by ∞ 9 2 γCDF = arg min (Fγ (z) FZ (z)) dz. (33) γ>0 0 − 8 h(γ) Z In Table I some numerical values of γCDF are given and 7 a comparison between three methods of shape parameter

6 estimation is available too. An overview from Table I shows that the optimal value of

C 5 γTail is near the value given by the kurtosis for any value 4 of n. It is clear also that γTail approaches closely to γKurt specially for n =2 for all values of δ. This analysis conﬁrms 3 the use of the kurtosis to approximate the PDF of the sum

2 of two independent GGRV by another GGD to get a good tail approximation, this may also conﬁrms that the kurtosis 1 0 1 2 3 4 5 6 7 8 9 10 γ measure the heavy tail. Unlike this observation, the γ obtained by minimizing the CDF error is a little bit far from outcomes Fig. 1: The curve of h(γ) for positive values of γ. of the kurtosis method. To conclude, these three methods can be used according to the situation we are facing. 6

TABLE I: Shape parameter for the approximated PDF using kurtosis, minimum CDF error, and minimum tail error for 1 0.95 σ1 =1 and different values of (α, β, δ) Optimal CDF 0.9 Kurtosis Optimal Tail γTail Simulation (α, β, δ) γKurt γCDF 0.85 n =0 1 2 3 Exact CDF . , . , . . . . . 0.8 (0 5 0 5 1) 0 626 0.467 0 768 0 673 0 624 0 642 (z) (0.5, 0.5, 2) 0.604 0.492 0.762 0.656 0.603 0.584 Z 0.75 (0.5, 0.7, 2) 0.633 0.501 0.861 0.741 0.636 0.834

(0.5, 1.2, 1) 0.779 0.602 1.160 1.053 0.757 1.165 CDF of Z, F 0.7 (1.5, 1.5, 2) 1.673 1.373 1.738 1.702 1.683 1.664 (1.5, 2.5, 1) 1.908 1.391 1.979 1.959 1.952 1.887 0.65 (1.5, 2.5, 2) 1.753 1.443 1.842 1.799 1.771 1.741 0.6 (2.5, 3 , 3) 2.295 1.941 2.226 2.261 2.267 2.335 0.55

0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 z 0.5 Simulation Exact PDF 0.45 Kurtosis Fig. 3: CDF of the sum of two GGRV using the Kurtosis, Optimal CDF 0.4 optimal tail, and optimal CDF approximations methods, for

0.35 α =2.5, β =1.5, δ =2, and σ1 =1 α=2.5, β=1.5, µ=0 α=0.5, β=1.5, µ=5

0.3

0.25

PDF of Z from the kurtosis method. Another observation is that all the 0.2 methods are close to each other and close to the CDF at the 0.15 saturation region, i.e. FZ (z) 1. This result is more detailed 0.1 in the next ﬁgure which shows≈ the complementary CDF in 0.05 Log scale.

0 −5 0 5 10 z

Simulation −1 10 Exact CCDF Fig. 2: Exact and approximated PDF of the sum of two GGRV, Optimal CDF Kurtosis −2 for β =1.5, δ =2, σ1 =1, and two values of α. 10 Optimal Tail

α=0.5, β=1.5 −3 10

−4 D. PDF and CDF Simulations 10

−5 The illustrations in this section are made for β =1.5, δ =2, 10 Complementary CDF of Z α β and σ =1. In Fig. 2, the PDF of the sum distribution is drawn −6 =2.5, =1.5 1 10 for two values of α (0.5 and 2.5) and µ takes two values to −7 split the curves of both cases. The exact and simulated PDF of 10

Z are drawn in the same figure among with the approximated −8 10 5 10 15 20 25 PDF. The latter is computed using the kurtosis and the optimal z CDF methods. It is clear that the exact PDF matchs perfectly the simulated PDF. Far from the mean, the approximated PDF Fig. 4: Complementary CDF of the sum of two GGRV for two appears close to the exact PDF and both methods have a good values of α. tail approximation. For α < 2, the kurtosis and the optimal CDF method are close to each other and match only the exact PDF at the tail with huge difference at the mean as mentioned In Fig. 4, the complementary CDF of the distribution of the sum is drawn for two values of ( and ). For in [10]. However, for α > 2, the kurtosis method presents a α 2.5 0.5 good approximation of the PDF even around the mean, while both cases, the approximated CCDF using the optimal CDF the optimal CDF method represents a good approximation of method matches the exact CCDF. However, as seen in Fig. 2, the CDF as it will be seen later. We omit the optimal tail for α< 2, the CCDF obtained from the kurtosis and optimal method here because it is close to the kurtosis method as tail methods is not too close to the exact CCDF. While, for shown in Table I. However one can draw it easily using the α > 1, they are close to each other and asymptotically close values available in Table I. to the exact CCDF. Our second illustration, highlighted in Fig. 3, consists of drawing the CDF of the sum for α = 2.5, β = 1.5, δ = 2, V. CONCLUSION and σ1 = 1 using all three methods to approximate the PDF The statistics of the distribution of the sum of two inde- in linear scale. It is noticed that the results obtained from the pendent GGRV were derived in closed form in terms of the optimal tail method (for n =3) are very close to those issued FHF and BFHF respectively, and the distribution of sum was 7 approximated by another GGD using three estimation methods [8] J. 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